Mean Cover Time Research Articles

On cover times of Markov chains

. We consider the cover time of a discrete-time homogenous Markov chain, which is the time needed by the Markov chain to visit all its states. We analyze both the distribution and the moments of the cover time, and we are interested in exact results instead of asymptotic values of the mean cover time, which are generally considered in the literature. We first obtain several general results on the hitting time and the cover time of a subset of the state space, both in terms of distribution and moments. These results are then applied to particular graphs, namely the generalized cycle graph, the complete graph, and the generalized path graph. They lead to recurrence or analytic relations for the distribution and the mean value of their cover times.

On the mean first passage time and mean cover time of a non-backtracking random walk on graphs

The non-backtracking random walk is known to have nicer properties than simple random walks. However, the effect of graph structure on the properties of the non-backtracking random walk has not been fully clarified. In this paper, we elucidate the mean first passage time of the non-backtracking random walk on two different types of graphs with different characteristics as well as the mean cover time on random regular graphs. Furthermore, we analyze the effects of the graph structure on the properties of the non-backtracking random walk using several numerical examples. Our findings include that the number of nodes in random regular graphs has a great impact on the mean first passage time of the non-backtracking random walk.

Minimization of spatial cover times for impaired self-avoiding random walks: the mirage effect

Self-avoidance is a common mechanism to improve the efficiency of a random walker for covering a spatial domain. However, how this efficiency decreases when self-avoidance is impaired or limited by other processes has remained largely unexplored. Here we provide a numerical study in regular lattices for the case when the self-avoiding signal left by a walker both (i) saturates after successive revisits to a site, and (ii) evaporates, or disappears, after some characteristic time. We surprisingly reveal that the mean cover time becomes minimum for intermediate values of the evaporation time, leading to the existence of a nontrivial optimum management of the self-avoiding signal. We show that this is a consequence of a complex dynamics arising from the interplay between signal evaporation and signal saturation, in which evaporation has the capacity of creating some sort of mirages (sites or regions that the walker see as unvisited, though in fact they are not) that enhance the searcher mobility, so contributing to a more efficient exploration of the lattice that counteracts the effects of signal saturation. Remarkably, we argue both through scaling arguments and from numerical results, that this mirage effect will become more and more significant as long as the domain size increases.

Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

AbstractNetworks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical (MH) organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that MH topologies may provide an optimal structure for certain dynamical processes. We revisit a MH network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erdős–Rényi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect.

Cover time for random walks on arbitrary complex networks.

We present an analytical method for computing the mean cover time of a discrete-time random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This quantity is particularly important for random search processes and target localization on network structures. Based on the global mean first-passage time of target nodes, we derive a method for computing the cumulative distribution function of the cover time based on first-passage time statistics. Our method is viable for networks on which random walks equilibrate quickly. We show that it can be applied successfully to various model and real-world networks. Our results reveal an intimate link between first-passage and cover time statistics and offer a computationally efficient way for estimating cover times in network-related applications.

Two-Dimensional Random Interlacements and Late Points for Random Walks

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.

Cover times of random searches

The first-passage time relates the efficiency of a search process, but fails to do so for searches in which several targets are sought. Now, the distribution of times required for a random search to visit all sites has been determined analytically. How long must one undertake a random search to visit all sites of a given domain? This time, known as the cover time1, is a key observable to quantify the efficiency of exhaustive searches, which require a complete exploration of an area and not only the discovery of a single target. Examples range from immune-system cells chasing pathogens2 to animals harvesting resources3,4, from robotic exploration for cleaning or demining to the task of improving search algorithms5. Despite its broad relevance, the cover time has remained elusive and so far explicit results have been scarce and mostly limited to regular random walks6,7,8,9. Here we determine the full distribution of the cover time for a broad range of random search processes, including Lévy strategies10,11,12,13,14, intermittent strategies4,15,16, persistent random walks17 and random walks on complex networks18, and reveal its universal features. We show that for all these examples the mean cover time can be minimized, and that the corresponding optimal strategies also minimize the mean search time for a single target, unambiguously pointing towards their robustness.

Mean cover time of one-dimensional persistent random walks.

The cover time is defined as the time needed for a random walker to visit every site of a confined domain. Here, we focus on persistent random walks, which provide a minimal model of random walks with short-range memory. We derive the exact expression of the mean cover time of a one-dimensional lattice by such a persistent random walk, both for periodic and reflecting boundary conditions.

Comparison of Multiple Random Walks Strategies for Searching Networks

We investigate diverse random-walk strategies for searching networks, especially multiple random walks (MRW). We use random walks on weighted networks to establish various models of single random walks and take the order statistics approach to study corresponding MRW, which can be a general framework for understanding random walks on networks. Multiple preferential random walks (MPRW) and multiple simple random walks (MSRW) are two special types of MRW. As search strategies, MPRW prefers high-degree nodes while MSRW searches for low-degree nodes more efficiently. We analyze the first passage time (FPT) of wandering walkers of MRW and give the corresponding formulas of probability distributions and moments, and the mean first passage time (MFPT) is included. We show the convergence of the MFPT of the first arriving walker and find the MFPT of the last arriving walker closely related with the mean cover time. Simulations confirm analytical predictions and deepen discussions. We use a small random network to test the FPT properties from different aspects. We also explore some practical search-related issues by MRW, such as detecting unknown shortest paths and avoiding poor routings on networks. Our results are of practical significance for realizing optimal routing and performing efficient search on complex networks.

On the Mean and Variance of Cover Times for Random Walks on Graphs

A method is described for calculating the mean cover time for a particle performing a simple random walk on the vertices of a finite connected graph. The method also yields the variance and generating function of the cover time. A computer program is available which utilises the approach to provide results for vertex symmetric graphs. Some examples are given.